Scale-invariant random geometry from mating of trees: A numerical study

Abstract

The search for scale-invariant random geometries is central to the asymptotic safety hypothesis for the Euclidean path integral in quantum gravity. In an attempt to uncover new universality classes of scaleinvariant random geometries that go beyond surface topology, we explore a generalization of the mating of trees approach introduced by Duplantier, Miller, and Sheffield. The latter provides an encoding of Liouville quantum gravity on the 2-sphere decorated by a certain random space-filling curve in terms of a twodimensional correlated Brownian motion, that can be viewed as describing a pair of random trees. The random geometry of Liouville quantum gravity can be conveniently studied and simulated numerically by discretizing the mating of trees using the mated-CRT maps of Gwynne, Miller, and Sheffield. Considering higher-dimensional correlated Brownian motions, one is naturally led to a sequence of nonplanar random graphs generalizing the mated-CRT maps that may belong to new universality classes of scale-invariant random geometries. We develop a numerical method to efficiently simulate these random graphs and explore their possible scaling limits through distance measurements, allowing us in particular to estimate Hausdorff dimensions in the two-and three-dimensional setting. In the two-dimensional case these estimates accurately reproduce previous known analytic and numerical results, while in the three-dimensional case they provide a first window on a potential three-parameter family of new scale-invariant random geometries.